Boundary-Value Problems in a Cut Plane

We present a survey of the theory of boundary-value problems in the plane with curvilinear cuts. The problem of linear conjugation, the Riemann-Hilbert problem, and the Riemann-Hilbert-Poincare problem are considered in detail both in the classical setting and for a cut plane, with an emphasis on problems with shifts. The main focus is on the solvability conditions and index formulas in various function classes. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.